bchgenpoly - Generator polynomial of BCH code

Syntax

genpoly = bchgenpoly(n,k) genpoly = bchgenpoly(n,k,prim_poly) [genpoly,t] = bchgenpoly(...)

Description

genpoly = bchgenpoly(n,k) returns the narrow-sense generator polynomial of a BCH code with codeword length n and message length k. The codeword length n must have the form 2^m-1 for some integer m. The output genpoly is a Galois row vector in GF(2) that represents the coefficients of the generator polynomial in order of descending powers. The narrow-sense generator polynomial is LCM[m_1(x), m_2(x), ..., m_2t(x)], where LCM is the least common multiple, m_i(x) is the minimum polynomial corresponding to αⁱ, α is a root of the default primitive polynomial for the field GF(n+1), and t is the error-correcting capability of the code.

Note Although the bchgenpoly function performs intermediate computations in GF(n+1), the final polynomial has binary coefficients. The output from bchgenpoly is a Galois vector in GF(2) rather than in GF(n+1).

genpoly = bchgenpoly(n,k,prim_poly) is the same as the syntax above, except that prim_poly specifies the primitive polynomial for GF(n+1) that has A as a root. prim_poly is an integer whose binary representation indicates the coefficients of the primitive polynomial. To use the default primitive polynomial for GF(n+1), set prim_poly to [].

[genpoly,t] = bchgenpoly(...) returns t, the error-correction capability of the code.

Examples

The results below show that a [15,11] BCH code can correct one error and has a generator polynomial X⁴ + X + 1.

m = 4;
n = 2^m-1; % Codeword length
k = 11; % Message length
% Get generator polynomial and error-correction capability.
[genpoly,t] = bchgenpoly(n,k)

The output is

genpoly = GF(2) array. 
 
Array elements = 
 
     1     0     0     1     1


t =

     1

Limitations

The maximum allowable value of n is 511.

References

[1] Peterson, W. Wesley, and E. J. Weldon, Jr., Error-Correcting Codes, 2nd ed., Cambridge, MA, MIT Press, 1972.

bchenc

bchnumerr

Free Early Verification Kit

Learn how to apply early verification to your development process through these technical resources.

Get technical kit

Trial software available

How much time do you spend on testing to ensure implementation meets system-level requirements?

Learn more